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Theorem gsumhashmul 30744
Description: Express a group sum by grouping by nonzero values. (Contributed by Thierry Arnoux, 22-Jun-2024.)
Hypotheses
Ref Expression
gsumhashmul.b 𝐵 = (Base‘𝐺)
gsumhashmul.z 0 = (0g𝐺)
gsumhashmul.x · = (.g𝐺)
gsumhashmul.g (𝜑𝐺 ∈ CMnd)
gsumhashmul.f (𝜑𝐹:𝐴𝐵)
gsumhashmul.1 (𝜑𝐹 finSupp 0 )
Assertion
Ref Expression
gsumhashmul (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥))))
Distinct variable groups:   𝑥, 0   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥
Allowed substitution hint:   · (𝑥)

Proof of Theorem gsumhashmul
Dummy variables 𝑡 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumhashmul.f . . . . . . 7 (𝜑𝐹:𝐴𝐵)
2 suppssdm 7830 . . . . . . . 8 (𝐹 supp 0 ) ⊆ dom 𝐹
32, 1fssdm 6508 . . . . . . 7 (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
41, 3feqresmpt 6713 . . . . . 6 (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑥)))
54oveq2d 7155 . . . . 5 (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑥))))
6 gsumhashmul.b . . . . . 6 𝐵 = (Base‘𝐺)
7 gsumhashmul.z . . . . . 6 0 = (0g𝐺)
8 gsumhashmul.g . . . . . 6 (𝜑𝐺 ∈ CMnd)
9 gsumhashmul.1 . . . . . . . 8 (𝜑𝐹 finSupp 0 )
10 relfsupp 8823 . . . . . . . . 9 Rel finSupp
1110brrelex1i 5576 . . . . . . . 8 (𝐹 finSupp 0𝐹 ∈ V)
129, 11syl 17 . . . . . . 7 (𝜑𝐹 ∈ V)
131ffnd 6492 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
1412, 13fndmexd 7601 . . . . . 6 (𝜑𝐴 ∈ V)
15 ssidd 3941 . . . . . 6 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
166, 7, 8, 14, 1, 15, 9gsumres 19029 . . . . 5 (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg 𝐹))
17 nfcv 2958 . . . . . 6 𝑥(𝐹‘(1st𝑧))
18 fveq2 6649 . . . . . 6 (𝑥 = (1st𝑧) → (𝐹𝑥) = (𝐹‘(1st𝑧)))
199fsuppimpd 8828 . . . . . 6 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
20 ssidd 3941 . . . . . 6 (𝜑𝐵𝐵)
211adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝐹:𝐴𝐵)
223sselda 3918 . . . . . . 7 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝑥𝐴)
2321, 22ffvelrnd 6833 . . . . . 6 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → (𝐹𝑥) ∈ 𝐵)
241ffund 6495 . . . . . . . . 9 (𝜑 → Fun 𝐹)
25 funrel 6345 . . . . . . . . 9 (Fun 𝐹 → Rel 𝐹)
26 reldif 5656 . . . . . . . . 9 (Rel 𝐹 → Rel (𝐹 ∖ (V × { 0 })))
2724, 25, 263syl 18 . . . . . . . 8 (𝜑 → Rel (𝐹 ∖ (V × { 0 })))
28 1stdm 7725 . . . . . . . 8 ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st𝑧) ∈ dom (𝐹 ∖ (V × { 0 })))
2927, 28sylan 583 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st𝑧) ∈ dom (𝐹 ∖ (V × { 0 })))
307fvexi 6663 . . . . . . . . . . . 12 0 ∈ V
3130a1i 11 . . . . . . . . . . 11 (𝜑0 ∈ V)
32 fressupp 30451 . . . . . . . . . . 11 ((Fun 𝐹𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
3324, 12, 31, 32syl3anc 1368 . . . . . . . . . 10 (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
3433dmeqd 5742 . . . . . . . . 9 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = dom (𝐹 ∖ (V × { 0 })))
352a1i 11 . . . . . . . . . 10 (𝜑 → (𝐹 supp 0 ) ⊆ dom 𝐹)
36 ssdmres 5845 . . . . . . . . . 10 ((𝐹 supp 0 ) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
3735, 36sylib 221 . . . . . . . . 9 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
3834, 37eqtr3d 2838 . . . . . . . 8 (𝜑 → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 ))
3938adantr 484 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 ))
4029, 39eleqtrd 2895 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st𝑧) ∈ (𝐹 supp 0 ))
4124funresd 6371 . . . . . . . . . . 11 (𝜑 → Fun (𝐹 ↾ (𝐹 supp 0 )))
4241adantr 484 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → Fun (𝐹 ↾ (𝐹 supp 0 )))
4337eleq2d 2878 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (𝐹 supp 0 )))
4443biimpar 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
45 simpr 488 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ (𝐹 supp 0 ))
4645fvresd 6669 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥))
47 funopfvb 6700 . . . . . . . . . . 11 ((Fun (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 ))))
4847biimpa 480 . . . . . . . . . 10 (((Fun (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥)) → ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 )))
4942, 44, 46, 48syl21anc 836 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 )))
5033adantr 484 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
5149, 50eleqtrd 2895 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ∖ (V × { 0 })))
52 eqeq2 2813 . . . . . . . . . . 11 (𝑣 = ⟨𝑥, (𝐹𝑥)⟩ → (𝑧 = 𝑣𝑧 = ⟨𝑥, (𝐹𝑥)⟩))
5352bibi2d 346 . . . . . . . . . 10 (𝑣 = ⟨𝑥, (𝐹𝑥)⟩ → ((𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣) ↔ (𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)))
5453ralbidv 3165 . . . . . . . . 9 (𝑣 = ⟨𝑥, (𝐹𝑥)⟩ → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)))
5554adantl 485 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑣 = ⟨𝑥, (𝐹𝑥)⟩) → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)))
56 fvexd 6664 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → (2nd𝑧) ∈ V)
5727ad3antrrr 729 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → Rel (𝐹 ∖ (V × { 0 })))
58 simplr 768 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
59 1st2nd 7724 . . . . . . . . . . . . . . . . 17 ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
6057, 58, 59syl2anc 587 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
61 opeq1 4766 . . . . . . . . . . . . . . . . 17 (𝑥 = (1st𝑧) → ⟨𝑥, (2nd𝑧)⟩ = ⟨(1st𝑧), (2nd𝑧)⟩)
6261adantl 485 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → ⟨𝑥, (2nd𝑧)⟩ = ⟨(1st𝑧), (2nd𝑧)⟩)
6360, 62eqtr4d 2839 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 = ⟨𝑥, (2nd𝑧)⟩)
64 difssd 4063 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ∖ (V × { 0 })) ⊆ 𝐹)
6564sselda 3918 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧𝐹)
6665adantr 484 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧𝐹)
6763, 66eqeltrrd 2894 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹)
6863, 67jca 515 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → (𝑧 = ⟨𝑥, (2nd𝑧)⟩ ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹))
69 opeq2 4768 . . . . . . . . . . . . . . . 16 (𝑦 = (2nd𝑧) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, (2nd𝑧)⟩)
7069eqeq2d 2812 . . . . . . . . . . . . . . 15 (𝑦 = (2nd𝑧) → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, (2nd𝑧)⟩))
7169eleq1d 2877 . . . . . . . . . . . . . . 15 (𝑦 = (2nd𝑧) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹))
7270, 71anbi12d 633 . . . . . . . . . . . . . 14 (𝑦 = (2nd𝑧) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (𝑧 = ⟨𝑥, (2nd𝑧)⟩ ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹)))
7356, 68, 72spcedv 3550 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
74 vex 3447 . . . . . . . . . . . . . 14 𝑥 ∈ V
7574elsnres 5862 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐹 ↾ {𝑥}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
7673, 75sylibr 237 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 ∈ (𝐹 ↾ {𝑥}))
7713ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝐹 Fn 𝐴)
7822ad2antrr 725 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑥𝐴)
79 fnressn 6901 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
8077, 78, 79syl2anc 587 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
8176, 80eleqtrd 2895 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 ∈ {⟨𝑥, (𝐹𝑥)⟩})
82 elsni 4545 . . . . . . . . . . 11 (𝑧 ∈ {⟨𝑥, (𝐹𝑥)⟩} → 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)
8381, 82syl 17 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)
84 simpr 488 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩) → 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)
8584fveq2d 6653 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩) → (1st𝑧) = (1st ‘⟨𝑥, (𝐹𝑥)⟩))
86 fvex 6662 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
8774, 86op1st 7683 . . . . . . . . . . 11 (1st ‘⟨𝑥, (𝐹𝑥)⟩) = 𝑥
8885, 87eqtr2di 2853 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩) → 𝑥 = (1st𝑧))
8983, 88impbida 800 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩))
9089ralrimiva 3152 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩))
9151, 55, 90rspcedvd 3577 . . . . . . 7 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ∃𝑣 ∈ (𝐹 ∖ (V × { 0 }))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣))
92 reu6 3668 . . . . . . 7 (∃!𝑧 ∈ (𝐹 ∖ (V × { 0 }))𝑥 = (1st𝑧) ↔ ∃𝑣 ∈ (𝐹 ∖ (V × { 0 }))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣))
9391, 92sylibr 237 . . . . . 6 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ∃!𝑧 ∈ (𝐹 ∖ (V × { 0 }))𝑥 = (1st𝑧))
9417, 6, 7, 18, 8, 19, 20, 23, 40, 93gsummptf1o 19079 . . . . 5 (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑥))) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧)))))
955, 16, 943eqtr3d 2844 . . . 4 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧)))))
96 simpr 488 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
9796eldifad 3896 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧𝐹)
98 funfv1st2nd 7731 . . . . . . 7 ((Fun 𝐹𝑧𝐹) → (𝐹‘(1st𝑧)) = (2nd𝑧))
9924, 97, 98syl2an2r 684 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝐹‘(1st𝑧)) = (2nd𝑧))
10099mpteq2dva 5128 . . . . 5 (𝜑 → (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧))) = (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧)))
101100oveq2d 7155 . . . 4 (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧)))) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧))))
10295, 101eqtrd 2836 . . 3 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧))))
103 nfcv 2958 . . . 4 𝑧(1st𝑡)
104 fvex 6662 . . . . 5 (2nd𝑡) ∈ V
105 fvex 6662 . . . . 5 (1st𝑡) ∈ V
106104, 105op2ndd 7686 . . . 4 (𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ → (2nd𝑧) = (1st𝑡))
107 resfnfinfin 8792 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝐹 supp 0 ) ∈ Fin) → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
10813, 19, 107syl2anc 587 . . . . 5 (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
10933, 108eqeltrrd 2894 . . . 4 (𝜑 → (𝐹 ∖ (V × { 0 })) ∈ Fin)
11033rneqd 5776 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = ran (𝐹 ∖ (V × { 0 })))
111 rnresss 5858 . . . . . 6 ran (𝐹 ↾ (𝐹 supp 0 )) ⊆ ran 𝐹
1121frnd 6498 . . . . . 6 (𝜑 → ran 𝐹𝐵)
113111, 112sstrid 3929 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵)
114110, 113eqsstrrd 3957 . . . 4 (𝜑 → ran (𝐹 ∖ (V × { 0 })) ⊆ 𝐵)
115 2ndrn 7726 . . . . 5 ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (2nd𝑧) ∈ ran (𝐹 ∖ (V × { 0 })))
11627, 115sylan 583 . . . 4 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (2nd𝑧) ∈ ran (𝐹 ∖ (V × { 0 })))
117 relcnv 5938 . . . . . . . 8 Rel 𝐹
118 reldif 5656 . . . . . . . 8 (Rel 𝐹 → Rel (𝐹 ∖ ({ 0 } × V)))
119117, 118mp1i 13 . . . . . . 7 (𝜑 → Rel (𝐹 ∖ ({ 0 } × V)))
120 1st2nd 7724 . . . . . . 7 ((Rel (𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → 𝑡 = ⟨(1st𝑡), (2nd𝑡)⟩)
121119, 120sylan 583 . . . . . 6 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → 𝑡 = ⟨(1st𝑡), (2nd𝑡)⟩)
122 cnvdif 5973 . . . . . . . . . 10 (𝐹 ∖ (V × { 0 })) = (𝐹(V × { 0 }))
123 cnvxp 5985 . . . . . . . . . . 11 (V × { 0 }) = ({ 0 } × V)
124123difeq2i 4050 . . . . . . . . . 10 (𝐹(V × { 0 })) = (𝐹 ∖ ({ 0 } × V))
125122, 124eqtri 2824 . . . . . . . . 9 (𝐹 ∖ (V × { 0 })) = (𝐹 ∖ ({ 0 } × V))
126125eqimss2i 3977 . . . . . . . 8 (𝐹 ∖ ({ 0 } × V)) ⊆ (𝐹 ∖ (V × { 0 }))
127126a1i 11 . . . . . . 7 (𝜑 → (𝐹 ∖ ({ 0 } × V)) ⊆ (𝐹 ∖ (V × { 0 })))
128127sselda 3918 . . . . . 6 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → 𝑡(𝐹 ∖ (V × { 0 })))
129121, 128eqeltrrd 2894 . . . . 5 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
130105, 104opelcnv 5720 . . . . 5 (⟨(1st𝑡), (2nd𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })) ↔ ⟨(2nd𝑡), (1st𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
131129, 130sylib 221 . . . 4 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → ⟨(2nd𝑡), (1st𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
13227adantr 484 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → Rel (𝐹 ∖ (V × { 0 })))
133 eqidd 2802 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → {𝑧} = {𝑧})
134 cnvf1olem 7792 . . . . . . . . 9 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ {𝑧} = {𝑧})) → ( {𝑧} ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑧 = { {𝑧}}))
135134simpld 498 . . . . . . . 8 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ {𝑧} = {𝑧})) → {𝑧} ∈ (𝐹 ∖ (V × { 0 })))
136132, 96, 133, 135syl12anc 835 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → {𝑧} ∈ (𝐹 ∖ (V × { 0 })))
137136, 125eleqtrdi 2903 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → {𝑧} ∈ (𝐹 ∖ ({ 0 } × V)))
138 eqeq2 2813 . . . . . . . . 9 (𝑢 = {𝑧} → (𝑡 = 𝑢𝑡 = {𝑧}))
139138bibi2d 346 . . . . . . . 8 (𝑢 = {𝑧} → ((𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ (𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧})))
140139ralbidv 3165 . . . . . . 7 (𝑢 = {𝑧} → (∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ ∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧})))
141140adantl 485 . . . . . 6 (((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑢 = {𝑧}) → (∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ ∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧})))
142117, 118mp1i 13 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → Rel (𝐹 ∖ ({ 0 } × V)))
143 simplr 768 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑡 ∈ (𝐹 ∖ ({ 0 } × V)))
144 simpr 488 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩)
145 df-rel 5530 . . . . . . . . . . . . . 14 (Rel (𝐹 ∖ ({ 0 } × V)) ↔ (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
146119, 145sylib 221 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
147146ad3antrrr 729 . . . . . . . . . . . 12 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
148147, 143sseldd 3919 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑡 ∈ (V × V))
149 2nd1st 7723 . . . . . . . . . . 11 (𝑡 ∈ (V × V) → {𝑡} = ⟨(2nd𝑡), (1st𝑡)⟩)
150148, 149syl 17 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → {𝑡} = ⟨(2nd𝑡), (1st𝑡)⟩)
151144, 150eqtr4d 2839 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑧 = {𝑡})
152 cnvf1olem 7792 . . . . . . . . . 10 ((Rel (𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = {𝑡})) → (𝑧(𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 = {𝑧}))
153152simprd 499 . . . . . . . . 9 ((Rel (𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = {𝑡})) → 𝑡 = {𝑧})
154142, 143, 151, 153syl12anc 835 . . . . . . . 8 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑡 = {𝑧})
15527ad3antrrr 729 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → Rel (𝐹 ∖ (V × { 0 })))
15696ad2antrr 725 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
157 simpr 488 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑡 = {𝑧})
158 cnvf1olem 7792 . . . . . . . . . . 11 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑡 = {𝑧})) → (𝑡(𝐹 ∖ (V × { 0 })) ∧ 𝑧 = {𝑡}))
159158simprd 499 . . . . . . . . . 10 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑡 = {𝑧})) → 𝑧 = {𝑡})
160155, 156, 157, 159syl12anc 835 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑧 = {𝑡})
161146ad3antrrr 729 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
162 simplr 768 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑡 ∈ (𝐹 ∖ ({ 0 } × V)))
163161, 162sseldd 3919 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑡 ∈ (V × V))
164163, 149syl 17 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → {𝑡} = ⟨(2nd𝑡), (1st𝑡)⟩)
165160, 164eqtrd 2836 . . . . . . . 8 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩)
166154, 165impbida 800 . . . . . . 7 (((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → (𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧}))
167166ralrimiva 3152 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧}))
168137, 141, 167rspcedvd 3577 . . . . 5 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃𝑢 ∈ (𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢))
169 reu6 3668 . . . . 5 (∃!𝑡 ∈ (𝐹 ∖ ({ 0 } × V))𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ ∃𝑢 ∈ (𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢))
170168, 169sylibr 237 . . . 4 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃!𝑡 ∈ (𝐹 ∖ ({ 0 } × V))𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩)
171103, 6, 7, 106, 8, 109, 114, 116, 131, 170gsummptf1o 19079 . . 3 (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧))) = (𝐺 Σg (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡))))
172 fveq2 6649 . . . . . 6 (𝑡 = 𝑧 → (1st𝑡) = (1st𝑧))
173172cbvmptv 5136 . . . . 5 (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡)) = (𝑧 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑧))
17433cnveqd 5714 . . . . . . 7 (𝜑(𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
175174, 125eqtr2di 2853 . . . . . 6 (𝜑 → (𝐹 ∖ ({ 0 } × V)) = (𝐹 ↾ (𝐹 supp 0 )))
176175mpteq1d 5122 . . . . 5 (𝜑 → (𝑧 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑧)) = (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧)))
177173, 176syl5eq 2848 . . . 4 (𝜑 → (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡)) = (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧)))
178177oveq2d 7155 . . 3 (𝜑 → (𝐺 Σg (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡))) = (𝐺 Σg (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧))))
179102, 171, 1783eqtrd 2840 . 2 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧))))
180 nfcv 2958 . . 3 𝑦(1st𝑧)
181 nfv 1915 . . 3 𝑥𝜑
182 vex 3447 . . . 4 𝑦 ∈ V
18374, 182op1std 7685 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
184 relcnv 5938 . . . 4 Rel (𝐹 ↾ (𝐹 supp 0 ))
185184a1i 11 . . 3 (𝜑 → Rel (𝐹 ↾ (𝐹 supp 0 )))
186 cnvfi 8794 . . . 4 ((𝐹 ↾ (𝐹 supp 0 )) ∈ Fin → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
187108, 186syl 17 . . 3 (𝜑(𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
188112adantr 484 . . . 4 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → ran 𝐹𝐵)
189184a1i 11 . . . . . . 7 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → Rel (𝐹 ↾ (𝐹 supp 0 )))
190 simpr 488 . . . . . . 7 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → 𝑧(𝐹 ↾ (𝐹 supp 0 )))
191 1stdm 7725 . . . . . . 7 ((Rel (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
192189, 190, 191syl2anc 587 . . . . . 6 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
193 df-rn 5534 . . . . . 6 ran (𝐹 ↾ (𝐹 supp 0 )) = dom (𝐹 ↾ (𝐹 supp 0 ))
194192, 193eleqtrrdi 2904 . . . . 5 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ ran (𝐹 ↾ (𝐹 supp 0 )))
195111, 194sseldi 3916 . . . 4 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ ran 𝐹)
196188, 195sseldd 3919 . . 3 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ 𝐵)
197180, 181, 6, 183, 185, 187, 8, 196gsummpt2d 30737 . 2 (𝜑 → (𝐺 Σg (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧))) = (𝐺 Σg (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)))))
198 df-ima 5536 . . . . . . 7 (𝐹 “ (𝐹 supp 0 )) = ran (𝐹 ↾ (𝐹 supp 0 ))
199 supppreima 30454 . . . . . . . . 9 ((Fun 𝐹𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (ran 𝐹 ∖ { 0 })))
20024, 12, 31, 199syl3anc 1368 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (ran 𝐹 ∖ { 0 })))
201200imaeq2d 5900 . . . . . . 7 (𝜑 → (𝐹 “ (𝐹 supp 0 )) = (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))))
202198, 201syl5eqr 2850 . . . . . 6 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))))
203 funimacnv 6409 . . . . . . 7 (Fun 𝐹 → (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹))
20424, 203syl 17 . . . . . 6 (𝜑 → (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹))
205 difssd 4063 . . . . . . 7 (𝜑 → (ran 𝐹 ∖ { 0 }) ⊆ ran 𝐹)
206 df-ss 3901 . . . . . . 7 ((ran 𝐹 ∖ { 0 }) ⊆ ran 𝐹 ↔ ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 }))
207205, 206sylib 221 . . . . . 6 (𝜑 → ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 }))
208202, 204, 2073eqtrd 2840 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
209193, 208syl5eqr 2850 . . . 4 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
2108cmnmndd 18924 . . . . . . 7 (𝜑𝐺 ∈ Mnd)
211210adantr 484 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
212108adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
213 imafi2 30476 . . . . . . 7 ((𝐹 ↾ (𝐹 supp 0 )) ∈ Fin → ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin)
214212, 186, 2133syl 18 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin)
215193, 113eqsstrrid 3967 . . . . . . 7 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵)
216215sselda 3918 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → 𝑥𝐵)
217 gsumhashmul.x . . . . . . 7 · = (.g𝐺)
2186, 217gsumconst 19050 . . . . . 6 ((𝐺 ∈ Mnd ∧ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin ∧ 𝑥𝐵) → (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
219211, 214, 216, 218syl3anc 1368 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
220 cnvresima 6058 . . . . . . . 8 ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) = ((𝐹 “ {𝑥}) ∩ (𝐹 supp 0 ))
221209eleq2d 2878 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (ran 𝐹 ∖ { 0 })))
222221biimpa 480 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → 𝑥 ∈ (ran 𝐹 ∖ { 0 }))
223222snssd 4705 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → {𝑥} ⊆ (ran 𝐹 ∖ { 0 }))
224 sspreima 30409 . . . . . . . . . . 11 ((Fun 𝐹 ∧ {𝑥} ⊆ (ran 𝐹 ∖ { 0 })) → (𝐹 “ {𝑥}) ⊆ (𝐹 “ (ran 𝐹 ∖ { 0 })))
22524, 223, 224syl2an2r 684 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 “ {𝑥}) ⊆ (𝐹 “ (ran 𝐹 ∖ { 0 })))
226200adantr 484 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 supp 0 ) = (𝐹 “ (ran 𝐹 ∖ { 0 })))
227225, 226sseqtrrd 3959 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 ))
228 df-ss 3901 . . . . . . . . 9 ((𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 ) ↔ ((𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (𝐹 “ {𝑥}))
229227, 228sylib 221 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → ((𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (𝐹 “ {𝑥}))
230220, 229syl5req 2849 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 “ {𝑥}) = ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}))
231230fveq2d 6653 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (♯‘(𝐹 “ {𝑥})) = (♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})))
232231oveq1d 7154 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → ((♯‘(𝐹 “ {𝑥})) · 𝑥) = ((♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
233219, 232eqtr4d 2839 . . . 4 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘(𝐹 “ {𝑥})) · 𝑥))
234209, 233mpteq12dva 5117 . . 3 (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥))) = (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥)))
235234oveq2d 7155 . 2 (𝜑 → (𝐺 Σg (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)))) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥))))
236179, 197, 2353eqtrd 2840 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2112  wral 3109  wrex 3110  ∃!wreu 3111  Vcvv 3444  cdif 3881  cin 3883  wss 3884  {csn 4528  cop 4534   cuni 4803   class class class wbr 5033  cmpt 5113   × cxp 5521  ccnv 5522  dom cdm 5523  ran crn 5524  cres 5525  cima 5526  Rel wrel 5528  Fun wfun 6322   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674   supp csupp 7817  Fincfn 8496   finSupp cfsupp 8821  chash 13690  Basecbs 16478  0gc0g 16708   Σg cgsu 16709  Mndcmnd 17906  .gcmg 18219  CMndccmn 18901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-seq 13369  df-hash 13691  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-0g 16710  df-gsum 16711  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-submnd 17952  df-mulg 18220  df-cntz 18442  df-cmn 18903
This theorem is referenced by:  elrspunidl  31017
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